Timeline for How Can I Tell when A Subgroup of a Lie Group is Generated by Unipotents?
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| Apr 27, 2010 at 15:02 | vote | accept | Sam Ruth | ||
| Apr 26, 2010 at 18:35 | comment | added | Jim Humphreys | Short answer: no. Books by Borel, Springer, me have limited coverage of structure over arbitrary fields. Try AMS online surveys by Tits (1965) PSPUM/9, Springer (1979) PSPUM/33.1: e-math.ams.org/publications/online-books/online_subject (and note Springer's 3.4 for your original question). Extensive details are in papers by Borel-Tits (IHES papers online at www.numdam.org): Groupes reductifs (Publ. Math. IHES 27, 1965), etc. Lie groups are just an example there. But books by Margulis, Zimmer focus more on ergodic theory, less on background. Good luck! | |
| Apr 26, 2010 at 14:33 | comment | added | Sam Ruth | Thanks Jim. Is there a good book on algebraic groups and Borel-Tits structure theory? I'm ultimately interested in doing ergodic theory, if that helps narrow it down. | |
| Apr 25, 2010 at 22:30 | comment | added | Jim Humphreys |
The structure of arbitrary real semisimple Lie groups is complicated to develop (while "unipotent" elements are more visible for linear algebraic groups), so your first question takes work to sort out. For compact Lie groups the special feature is that every element is conjugate to one in a fixed maximal torus, hence semisimple. In the noncompact case, generation by unipotents appears concretely for classical groups like SO$(n,1)$; but in general Borel-Tits structure theory for algebraic groups is probably most helpful and covers all fields of definition: noncompact becomes "isotropic".
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| Apr 25, 2010 at 19:37 | comment | added | Sam Ruth | Why does non-compact mean that it must have non-trivial unipotents? And does the subgroup they generate being closed and normal imply it has to be all of $So(2,1)$? I don't know much about the general structure of Lie groups-everything I know comes from Fulton & Harris. I know even less about algebraic groups. | |
| Apr 25, 2010 at 17:54 | history | edited | Jim Humphreys | CC BY-SA 2.5 |
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| Apr 24, 2010 at 22:11 | history | answered | Jim Humphreys | CC BY-SA 2.5 |